Introduction
In this project we present a qualitative analysis of the motion of a
hydroplane racing boat like the one shown in the image above. In this model we will
focus only on its forward movement and will ignore any pitching, rolling and movement in
other directions. The model is constructed using the basic force balancing
equation. In particular, the total force is equal to the thrust generated by
the motor minus the drag due to water and air resistance. This gives the
differential equation
where v is the velocity, m is the mass and T(v) and W(v) are functions describing the thrust and drag respectively.
We will assume that the thrust is constant i.e. T(v) = T. On the other hand, the drag should be zero at zero velocity and initially increase with increasing velocity. However, because the boat rises in the water as v increases, the amount of water per unit surface area decreases and thus the drag will decrease for larger velocities. As v is increased still further the drag will again increase due primarily to higher air resistance. In other words, the graph of the function W(v) is qualitatively cubic in shape.
In the exercises that follow you will examine the consequences of this drag assumption
on the motion of the hydroplane.
Problem 1. The assumption that the thrust T
is constant gives the differential equation
mv' = T - W(v).
Show that the critical points of this differential equation are given by T = W(v). The left side of the applet below plots v on the horizontal axis and T on the vertical axis. The yellow curve is the graph of the critical points. Use this applet to describe the equilibrium velocities for various values of T. In particular, how and why does the equilibrium velocity depend on the initial velocity?
Problem 2. Suppose the hydroplane is initially
at rest so that T(0)=0. What happens as the thrust T is slowly increased to
a high value of T? To investigate this situation we use an additional differential
equation to describe the changing thrust. In particular, let's assume that the rate of
change of thrust is constant so we get the system of differential equations
mv' = T - W(v)
T' = a
where a is a constant (In the applet below a=0.07). Use the applet below to explain this phenomenon.
Problem 3. Now that the hydroplane is up to
speed the driver must bring the boat to a stop after passing the finish line. By changing
the sign of a in the applet above, describe how the velocity changes as the thrust
is decreased.
The Unlimited Hydroplane Racing Association
Homepage
The IDEA site is undergoing renovations, in conjunction with the renewed
activities of CODEE (as described below). Thank you for your patience.
CODEE, the Consortium for Differential Equations Experiments, has been
revitalized. CODEE was quite active in the 1990s in spreading differential
equations activities, information, and software tools. In particular,
CODEE formed the organization for the ODE Architect software. Recently,
an NSF project headed by Darryl Young of Harvey Mudd College has
reinvigorated CODEE.